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The dot product Hessian of the spheroid

The second partial derivative of the single spheroidal dot product δz with respect to the orientational parameters $\mathfrak{O}_i$ and $\mathfrak{O}_j$ is

$\displaystyle {\frac{{\partial^2 \delta_z}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$ = $\displaystyle {\frac{{\partial^2}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$$\displaystyle \left(\vphantom{ \widehat{XH} \cdot \widehat{\mathfrak{D}_{\scriptscriptstyle \parallel}} }\right.$$\displaystyle \widehat{{XH}}$⋅$\displaystyle \widehat{{\mathfrak{D}_{\scriptscriptstyle \parallel}}}$$\displaystyle \left.\vphantom{ \widehat{XH} \cdot \widehat{\mathfrak{D}_{\scriptscriptstyle \parallel}} }\right)$
= $\displaystyle \widehat{{XH}}$$\displaystyle {\frac{{\partial^2 \widehat{\mathfrak{D}_{\scriptscriptstyle \parallel}}}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$.		 (15.186)

The Dpar Hessian

The second partial derivatives of the unit vector $\widehat{{\mathfrak{D}_{\scriptscriptstyle \parallel}}}$ with respect to the spherical angles are
\begin{subequations}\begin{align} \frac{\partial^2 \widehat{\mathfrak{D}_{\scrip... ...\ -\sin \theta \sin \phi \\ 0 \\ \end{pmatrix}. \end{align}\end{subequations}


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